Momentum map reduction for nonholonomic systems

TL;DR

This paper develops a two-step reduction method for nonholonomic systems with symmetries, generalizing existing constructions and connecting to almost symplectic structures and reduced brackets, with applications to classical examples.

Contribution

It introduces a novel two-step reduction procedure for nonholonomic systems that extends Marsden-Weinstein reduction to almost symplectic manifolds.

Findings

The second step yields almost symplectic leaves matching reduced nonholonomic brackets.

The procedure generalizes previous reduction methods for nonholonomic systems.

Classical examples demonstrate the effectiveness of the proposed reduction.

Abstract

This paper presents a reduction procedure for nonholonomic systems admitting suitable types of symmetries and conserved quantities. The full procedure contains two steps. The first (simple) step results in a Chaplygin system, described by an almost symplectic structure, carrying additional symmetries. The focus of this paper is on the second step, which consists of a Marsden-Weinstein--type reduction that generalizes constructions in [4,17]. The almost symplectic manifolds obtained in the second step are proven to coincide with the leaves of the reduced nonholonomic brackets defined in [7]. We illustrate our construction with several classical examples.